# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=5#results)[1](/en/knowledge/?page=1#results) … [5](/en/knowledge/?page=5#results) 6 [7](/en/knowledge/?page=7#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=7#results) 302 results · Page 6 of 13 Knowledge object ## [Discrete-time stochastic process](/en/mko/mko-discrete-stochastic-process/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law. mko-discrete-stochastic-process · Probability and statistics Knowledge object ## [Multivariable real-valued function](/en/mko/mko-multivariable-real-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data. mko-multivariable-real-function · Analysis Knowledge object ## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space. mko-linear-independence-basis · Algebra Knowledge object ## [Measurable function](/en/mko/mko-measurable-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra. mko-measurable-function · Analysis Knowledge object ## [Finite-state Markov chain](/en/mko/mko-finite-markov-chain/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities. mko-finite-markov-chain · Probability and statistics Knowledge object ## [Partial derivative](/en/mko/mko-partial-derivative/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero. mko-partial-derivative · Analysis Knowledge object ## [Finite convex combination](/en/mko/mko-finite-convex-combination/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one. mko-finite-convex-combination · Geometry Knowledge object ## [Partial differential equation](/en/mko/mko-partial-differential-equation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention. mko-partial-differential-equation · Differential equations Knowledge object ## [Convex set](/en/mko/mko-convex-set/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set. mko-convex-set · Geometry Knowledge object ## [Boundary-value problem](/en/mko/mko-boundary-value-problem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness. mko-boundary-value-problem · Differential equations Knowledge object ## [Convex function](/en/mko/mko-convex-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values. mko-convex-function · Analysis Knowledge object ## [Convex optimization problem](/en/mko/mko-convex-optimization-problem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations. mko-convex-optimization-problem · Optimization and operations research Knowledge object ## [Finite abelian group](/en/mko/mko-finite-abelian-group/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements. mko-finite-abelian-group · Algebra Knowledge object ## [Real normed vector space](/en/mko/mko-real-normed-vector-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality. mko-real-normed-vector-space · Analysis Knowledge object ## [Cauchy sequence](/en/mko/mko-cauchy-sequence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart. mko-cauchy-sequence · Analysis Knowledge object ## [Finite joint distribution](/en/mko/mko-finite-joint-distribution/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution. mko-finite-joint-distribution · Probability and statistics Knowledge object ## [Covariance](/en/mko/mko-covariance/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence. mko-covariance · Probability and statistics Knowledge object ## [Graph isomorphism](/en/mko/mko-graph-isomorphism/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair. mko-graph-isomorphism · Combinatorics and discrete mathematics Knowledge object ## [Equivalence relation](/en/mko/mko-equivalence-relation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity. mko-equivalence-relation · Foundations and logic Knowledge object ## [Finite unital ring](/en/mko/mko-finite-unital-ring/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero. mko-finite-unital-ring · Algebra Knowledge object ## [Norm-induced metric](/en/mko/mko-norm-induced-metric/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms. mko-norm-induced-metric · Analysis Knowledge object ## [Complete metric space](/en/mko/mko-complete-metric-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A metric space is complete when every Cauchy sequence converges to a point still belonging to that space. mko-complete-metric-space · Analysis Knowledge object ## [Real bilinear form](/en/mko/mko-real-bilinear-form/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input. mko-real-bilinear-form · Algebra Knowledge object ## [Linear map between real vector spaces](/en/mko/mko-general-linear-map/?from=%2Fen%2Fknowledge%2F%3Fpage%3D6) A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars. mko-general-linear-map · Algebra [← Previous](/en/knowledge/?page=5#results)[1](/en/knowledge/?page=1#results) … [5](/en/knowledge/?page=5#results) 6 [7](/en/knowledge/?page=7#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=7#results) 302 results · Page 6 of 13